Is there any theorem giving the maximum number of limit cycles that a system can have? More specifically, my system is
$\dot{x} = -r(x+c)(x+y+z)$
$\dot{y} = b_1(a_1 x + (1-a_1)(z-y))$
$\dot{z} = b_2(a_2 x + (1-a_2)(y-z))$
where $b_1,b_2,r\in(0,+\infty)$ and $a_1,a_2,c\in(0,1)$ are fixed parameters. $x$ is bounded in $(-c,+\infty)$
which has a unique equilibrium at the origin, which is locally stable in a large region of the parameter space. Can something be said about the maximum number of limit cycles for this system?